User:Benbella/Surface Water Quality Modeling:Physical Process

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Contents 1 Geometric Representation 1.1 Zero-Dimensional Models 1.2 One-Dimensional Models 1.3 Multi-Dimensional Models 2 Temporal Variation 3 Adjective Transport 3.1 Empirical Specification of Advection 3.2 Zero Dimensional Models for Lakes 3.3 One-Dimensional Models for Lakes 3.4 One-Dimensional Models for Rivers 3.5 One-Dimensional and Pseudo-Two-Dimensional Models for Estuaries 3.6 Two-Dimensional Vertically Averaged Models for Lakes and Estuaries 3.7 Two-Dimensional Laterally Averaged Models for Reservoirs and Etuaries 3.8 Three-dimensional Models for Lakes and Estuaries 4 Dispersive Transport 4.1 Introduction 4.2 Vertical Dispersive Transport 4.3 Horizontal Eddy Diffusive Transport 4.4 Longjtudinal Dispersive Transport in Estuaries 4.4.1 Time Varying Longitudinal Dispersion 4.4.2 Steady State Longitudinal Dispersion 4.4.3 The Lagrangian Method 4.5 Dispersive Transport in Rivers 4.5.1 Introduction 4.6 Longitudinal Dispersion in Rivers 4.6.1 Lateral Dispersion In Rivers 4.7 Summary 5 Surface Heat Budget 5.1 Measurement Units 5.2 Net short wave Solar Radiation, Qsn 5.3 Net Atmospheric Radiation, Qan 5.4 Long Wave Radiation, Qbr 5.5 Evaporative Heat Flux, Qe 5.6 Convective Heat Flux, Qc 5.7 Equilibrium Temperature and Linearization 5.7.1 Equilibrium temperature, Te 5.7.2 Surface Heat Exchange Coefficient, K 5.8 Heat Exchange with Stream Bed 5.9 Summary 6 References

[edit] Geometric Representation

[edit] Zero-Dimensional Models

Zero—dimensional models are used to estimate spatially averaged pollutant concentrations at minimum cost. These models predict a concentration field of the form C = g(t), where t represents time. They cannot predict the fluid dynamics of a system, and the representation is usually such that an analytical solution is possible. As an example, the simplest representation of a lake is to consider it as a continuously stirred tank reactor (CSTR).

[edit] One-Dimensional Models

Most river models use a one-dimensional representation, where the system geometry is formulated conceptually as a linear network of segments or volume sections (see Figure 2-1). Variation of water quality parameters occur longitudinally (in the x-direction) as the water is transported out of one segment and into the next. The one—dimensional approach is also a popular method for simulation of small, deep lakes, where the vertical variation of temperature and other quality parameters is represented by a network of vertically stacked horizontal slices or volume segments.

[edit] Multi-Dimensional Models

Water quality models of lakes and estuaries are often two- or three- dimensional In order to represent the spatial heterogeneity of the water bodies. Depending on the system, two-dimensional representations include a vertical dimension with longitudinal segmentation for deep and narrow lakes, reservoirs, or estuaries (Figure 2-2). Three-dimensional spatial representations have been used to model overall lake circulation patterns. Part of the reason for this need is the concern with the water quality of the near-shore zone as well as deep zones of lakes. In addition, the different water quality interactions in these zones can lead to changes in the overall lake quality that cannot be predicted without this spatial definition.

[edit] Temporal Variation

Ecological models are distinguished on a temporal basis as being either "dynamic" or "steady-state". A strict steady-state assumption implies that the vatlables in the system equations do not change with time. Forcing functions, or exogenous variables, that describe environmental conditions which are unaffected by internal conditions of the system, have constant values. Inflows and outflows are discharged to and drawn from the system at

a constant rate and any other hydrologic phenomena are also steady. Insolation, light intensity, photoperiods, extinction coefficients, and settl ing rates are a few examples of additional forcing functions which are held constant in a steady-state model . Constant forcing functions represent mean conditions observed in a system, and therefore the model cannot simulate cyclic phenomena.

A wide variety of planning problems can be analyzed by use of steady- state or quasi-steady (slowly varying) mathematical models which provide the necessary spatial detail for important water quality variables. Certain phenomena can achieve steady-state conditions within a short time interval and therefore can be modeled rather easily. Steady-state or quasi-steady representations are particularly useful because of their simplicity. Examples of phenomena which have been modeled on a steady-state basis are:

1. bacterial die-off,
2. dissolved oxygen concentrations (under certain conditions), and
3. nutrient distribution and recycle.

Many water quality or ecological models for rivers and lakes are concerned with the simulation of water quality variables that have substantial temporal variation and are linke.d to processes and variables that vary considerably. For example, the seasonal distribution of certain biological species and related abiotic substances may be of major importance. In these instances, dynamic models are required.

The process of selecting the correct time and length scales and then matching these with an appropriate model demands both an a priori understanding of the dominate physical, chemical and biological processes occurring within the systeni, as well as an understanding of a given model’s theoretical basis and practical application limits. Proper guidance for model selection and application best comes from a thorough review of the relevant literature appropriate to the specific problem at hand. Ford and Thornton (1979), for example, present a detailed discussion of the time and length scales appropriate for the vertical one-dimensional modeling approach for reservoirs and lakes. The references presented in Table 2-1 as well as several others cited throughout this chapter discuss model compatibility requirements for various water body types and applications.

The remainder of this chapter focuses on advective transport, dispersive transport, and the surface heat budget.

[edit] Adjective Transport

The concentration of a substance at a particular site within a system is continually modified by the physical processes of advection and dispersion which transport fluid constituents from location to location. However, the total amount of a substance in a closed system remains constant unless it is modified by physical, chemical, or biological processes. Employing a Fickian type expression for turbulent mass flux, the threedimensional advection-diffusion (mass balance) equation can be written as:

Difficulties exist in trying to correctly quantify the terms in this equation. The unsteady velocity field (u,v,w) is usually evaluated separately from Equation (2-1) so that the pollutant concentration, c, can be prescribed. The complete evaluation of the velocity field involves the simultaneous solution of the momentum, continuity, hydrostatic, and state equations in three dimensions (see Leendertse and Liu, 1975; Hinwood and Wallis, 1975). Although sophisticated hydrodynamic models are available, the detail and expense of applying such models are often not justified in water quality computations, especially for long term or steady-state simulations where only average flow values are required. For example, the annual thermal cycle for a strongly stratified reservoir with a relatively low ir1flow to volume ratio has been successfully simulated with only a crude one-dimensional, steady-state application of mass and energy conservation principles. On the other hand, simulation of large, weakly stratified impoundments dominated by wind driven circulation may require the ultimate, full representation of the unsteady velocity field in three dimensions.

The purpose of this section is to briefly familiarize the reader with the various types of approaches used to evaluate the velocity field in water quality models. Most hydrodynamic models internally calculate hydrodynamics with relatively little user control except for specification of forcing conditions such as wind, tides, inflows, outflows and bottom friction. Thus, the following paragraphs present only a summary discussion of the approaches used, organized according to the dimensional treatment of the model.

[edit] Empirical Specification of Advection

This is the crudest approach, in that the advective terms of the advection-diffusion equation (Equation 2-1) are directly specified from field data. Empirical specification is quite common in water quality models for rivers, but is also often used in steady-state or slowly-varying estuary water quality models (e.g., O’Connor et al. (1973)). In these types of estuary models, specification of the dispersion coefficients is critical since dispersion must account for the mixing which in reality is caused by the oscillatory tidal action. Due to the highly empirical treatment of the physical processes in such models, the model "predictions" remain valid for only those conditions measured in the field. These models cannot predict water quality variations under other conditions, thus increasing the demand on field data requirements. Examples of models representative of the above approach include O’Connor et at. (1973) and Tetra Tech (1977).

[edit] Zero Dimensional Models for Lakes

A coarse representation of the water system as a continuously stirred tank reactor (CSTR) is often sufficient for problem applications to some lakes where detailed hydrodynamics are not required. Since in this zero— dimensional type of representation there is only a single element, no transport direction can be specified. The quantity of flow entering and leaving the system alone determines water volume changes within-the element. Examples of zero-dimensional models include lake models by Anderson et al. (1976).

[edit] One-Dimensional Models for Lakes

For lakesystenis with long residence times and stratification in the vertical direction, vertical one—dimensional representations are common. Horizontal layers are imposed •and advective transport is assumed to occur only in the vertical direction.

Generally the tributary inflows and outflows are assumed to enter and leave the lake at water levels of equal density. Since water is essentially incompressible the inflow is assumed to generate vertical advective flow (via the continuity equation) between all elements above the level of entry. The elements below this level, containing higher density water, are assumed to be unaffected. Examples of one-dimensional lake models include Lombardo (1972, 1973), Baca et al. (1974), Chen and Orlob (1975), Thomann et al. (1975), Imberger et al. (1977), HEC (1974), Markofsky and Harleman (1973), and CE-QUAL-R1(1982).

For lake or reservoir systems exhibiting complex horizontal interfiows, inflows, and outflows, seiii-empirical formulations have been developed to distribute inflows and tà determine the vertical location from which outflows arise, depending on stratification conditions. Examples include models by U.S. Army Corps of Engineers (1974), Baca et al. (1976), and Tetra Tech (1976).

[edit] One-Dimensional Models for Rivers

Most river models represent river systems conceptually as horizontal linear networks of segments or volume elements. The process of advection is assumed to transport a constituent horizontally by movement of the parcel of water containing the constituent. In general, there are two approaches to treat the advection in river models. One approach requires field calibration of the river flow properties by measuring flows and cross sectional geometry at each model segment over a range of flow magnitudes. A power series can then be developed for each cross section to interpolate or extrapolate for other flow events. Such an approach is especially appropriate for rivers exhibiting complex hydraulic properties (i.e., supercritical flows, cascades, etc.) and when steady state solutions are of interest. Examples of such models Include Tetra Tech (1977).

A second, more rigorous approach for simulating river advection involves the simultaneous solution of the continuity and momentum equations for the portion of the river under study. This approach is considered more "predictive" than the former since empirical flow data are required only for model calibration and verification. It is also more accurate and appropriate for use in transient water quality simulations. In either case, however, geometrical data on the cross-sectional shapes of the river are required. Examples of models representative of the latter approach include Brocard and Harleman (1976), and Peterson etal. (1973).

[edit] One-Dimensional and Pseudo-Two-Dimensional Models for Estuaries

A natural extension of the one—dimensional river model has been to estuary systems, either as a one-dimensional representation for narrow estuaries or as a system of multiple interconnecting one-dimensional channels for pseudo-representation of wider or multi-channeled estuaries. In either case, advection is determined through the simultaneâus solution of the continuity and momentum equations together with appropriate tidal boundary conditions. These types of models are generally quite flexible in their ability to handle multiple inflows, transient boundary conditions, and complex geometrical configurations. Two primary approaches include the "link—node" network models by Water Resources Engineers (WRE) (1972), and the finite element model (Galerkin Method) by Harleman et al. (1977).

[edit] Two-Dimensional Vertically Averaged Models for Lakes and Estuaries

Vertically averaged, two-dimensional models have proven to be quite useful, especially in modeling the hydrodynamics and water quality of relatively shallow estuaries and wind—driven lakes. The crucial assumption of these models is the vertically well-mixed layer that allows for vertiãal integration of the continuity, momentum, arid mass-transport equations. Such models are frequently employed to provide the horizontal advection for water quality models since they are relatively inexpensive to operate compared to the alternatives of large scale field measurement programs or fully threedimensional model treatments. There exist well over fifty models which would fit into the two-dimensional, vertically averaged classification. Examples of models that have been widely used and publicized include Wang and Connor (1975), Leendertse (1970), Taylor and Pagenkopf (1980), and Simons (1976).

[edit] Two-Dimensional Laterally Averaged Models for Reservoirs and Etuaries

In recent years, laterally averaged models have become standard simulation techniques for reservoirs or estuaries which exhibit significant vertical and longitudinal variations in density and water quality conditions. The two-dimensional laterally averaged models require the assumption of uniform lateral mixing in the .cross channel direction. Although this simplification eliminates one horizontal dimension, the solution of the motion equations in the remaining longitudinal and vertical dimensions requires a much more rigorous approach than for the two- dimensional vertically averaged models. In order to correctly simulate the vertical effects. of density gradients on the hydrodynamics and mass transport, both the motion (continuity and momentum) and advective-diffusion equations must be solved simultaneously. In addition, such models must also treat the vertical eddy viscosity (momentum transfer due to velocity gradients) and eddy diffusivity (mass transfer due to concentration gradients) coefficients, which are directly related to the degree of internal mixing and the density structure over the water column. Mathematical treatment of vertical diffusion and vertical momentum transfer varies greatly between models, and will be discussed further in this document. Examples of laterally averaged reservoir models include Edinger and Buchak (1979) and Norton et al. (1973). Examples of laterally averaged models developed for estuaries include Blumberg (1977), Najarian et al. (1982) and Wang (1979).

[edit] Three-dimensional Models for Lakes and Estuaries

Fully three-dimensional and layered models have been the subject of considerable attention over the last decade. Although still a developing field, there are a number of models which have been applied to estuary, ocean, and lake systems with moderate success. As with laterally averaged two-dimensional models, the main technical difficulty in this approach is in the specification of the internal turbulent momentum transfer and mass diffusivities, which are Ideally calibrated with field observations, thus making availability of adequate prototype data an important consideration. An additional factor of great importance is the relatively large computation cost of running three-dimensional models, especially for long-term water quality simulations. In many cases, the effort and cost of running such models is difficult to justify from purely a water quality standpoint. However, as computational costs continue to decrease and sophistication of numerical techniques increases, such models will eventually play an important role in supplying the large scale hydrodynamiC regimes in water quality simulations. Examples of the more prominent three—dimensional models include Blumberg and Mellor (1978), Leendertse and Liu (1975), Sheng and Butler (1982), Simons (1976) and King (1982).

[edit] Dispersive Transport

[edit] Introduction

The purpose of this section is to show how dispersive transport terms are incorporated into the equations of motion and continuity by temporal and spatial averaging (a detailed discussion of this subject is also given by Fischer et al. (1979)). A consequence of temporal averaging of either instantaneous velocity or concentration is to produce a smoothed velocity or. concentration response curve over time. Figure 2-3 illustrates both instantaneous velocity and time-smoothed curves. The velocities V and V are related by:

By averaging, the stochastic components are removed from the momentum and mass conservation equations. However, cross product terms appear in the equations, such as VV and VV in the case of the momentum equation, and VxC and VyC’ in the case of the mass conservation equation (where C’ is the instantaneous concentration fluctuation, and Vx and Vy are the random velocity deviations in the x and y directions, respectively). In the case of the momentum equation these terms are called turbulent momentum fluxes, and in the case of the mass conservation equation they are called turbulent mass fluxes. It is through these terms that eddy viscosity and eddy diffusivity enter into the momentum and mass conservation equations.

To solve the time-smoothed equations, the time averaged cross product terms are expressed as functions of time averaged variables. Numerous empirical expressions have been developed to do this. The expressions most often applied are analogous to Newton’s law of viscosity in the case of turbulent momentum transport and Fick’s law of diffusion in the case of turbulent mass transfer. Expressed quantitatively these relationships are of the form:

In natural water bodies the turbulent viscosity and diffusivity given by Equations (2-3) and (2—4) swamp their counterparts on the molecular level. The relative magnitude between eddy diffusivities and molecular diffusion coefficients is depicted graphically in Figure 2-4.

In addition to temporal averaging, spatial averaging is often used to simplify three dimensional models to two or one dimensions. As an illustration consider the vertically averaged mass transport equation. Before averaging, the governing three dimensional mass transport equation is typically written as:

Before spatial averaging, the local concentration and velocities can be expressed by a vertically averaged term and a deviation term:

It is noted that when vertical integration is performed on the three-dimensional mass conservation equation, cross product terms appear in the resulting two—dimensional equation, just as they do when temporal averaging is done because vertical gradients generally exist in both the concentration and velocity fields. The horizontal turbulent diffusion fluxes Qx, Qy are usually expressed in terms of the gradients of the vertically averaged concentration and the turbulent diffuion coefficient, which in general form are written:

One-dimensional mass conservation equations result when a second spatial averaging is performed. The one-dimensional equations express changes along the main flow axis. As expected, the diffusion terms are again different from their two-dimensional counterparts. Consequently, the type and magnitude of the diffusion terms appearing in the simulation equations depends not only on the water body characteristics, but the model used to simulate the water body.

[edit] Vertical Dispersive Transport

Vertical dispersive transport of momentum and mass becomes important in lakes or estuaries characterized by moderate to great depths. In a lake environment, vertical mixing is generally caused by wind action on the surface through which eddy turbulence is transmitted to the deeper layers by the action of shear stresses. In estuaries, typically the vertical mixing is induced by the Internal turbulence driven by the tidal flows, in addition to surface wind effects. Similarly, the internal mixing in deep reservoirs is primarily caused by the flow—through action. In each environment, however, the amount of vertical mixing is controlled, to a large extent, by the degree of density stratification in the water body. Treatment of vertical mixing processes in mathematical models is generally achieved through the specification of vertical eddy viscosity (Ev) and eddy dfffusivity (Ky) terms, as previously discussed. As observed by McCutcheon (1983), however, there is little consensus on what values the vertical eddy coefficients should have and how eddy viscosity and eddy diffusivity are related. At present, the procedure for estimating these coefficients is generally limited to empirical techniques that range from specifying a constant Ev and K to.relating to some measure of stability, i.e., the Richardson number Ri. In this approach, the ratio of the coefficients for stratified flow to the coefficients for unstratified flow is expressed as a function of stability f(s):

As reported by McCutcheon (1983), in a recent review of available data for stratified water flows (Deift, 1979) Equations. (2-16) and (2-17) were found to fit the data better than several other similar formulations. Models by Waidrop (1978), Harper and Waidrop (1981), Edinger and Buchak (1979), O’Connor and Lung (1981), Najarian et al. (1982), and Heinrich, Lick and Paul (1981) use this scheme. In some models, the coefficients and exponents in EquatIon (2—16) and (2—17) are not adjusted, and any discrepancies between field measurements and model predictions are attributed to the Inexactness of the model. In other models, the coefficients and exponents are calibrated on a site specific basis.

For model simulations of mixing through and below the thermocline, the Munk and Anderson type formulas appear to be less adequate (McCutcheon, 1983). Odd and Rodger (1978) developed site specific eddy viscosity formulations for the Great Ouse Estuary in Britain:

Image:Equation 2 19.jpg

where b and n are coefficients. The depth varying RI is used if Ri increases continuously starting at the bed and extending over 75 percent or more of the depth. Where a significant peak in Ri occurs inthe vertical gradient, that peak Ri is used for all depths in the equation above. McCormick and Scavia (1981) make a correction for Kv in Lake Ontario and Lake Washington studies that is similar to corrections of Ev by Odd and Rodger. Above the hypolimnion, they apply a modification of the Kent and Pritchard (1959) equation:

Below the thennocline a constant Kv was specified for Lake Ontario. In Lake Washington, Equation (2-21) and (2—22) were applied throughout. the depth. In Lake Washington bottom shear was important for mixing as opposed to deeper Lake Ontario where surface wind shear dominated the mixing process.

Several other formulations for Ev and Kv have been developed which are not based on the Munk and Anderson equations. For example, Blumberg (1977), in his laterally averaged model of the Potomac River Estuary, employed an expression for Ky which uses a ratio of Ri to a critical Ri to relate Ky to stability, where:

Simons also assumed that the vertical eddy viscosity coefficient was based on a similar relationship. The above formulation is a result of experiments performed in fjords, coastal and open sea areas, as well as from Lake Ontario, and is generally valid for expressing the vertical mixing In the upper 20 m for persistent winds above 4-5 m/sec. The lower value of the numerical constant refers to the lake data and the higher value to the oceanic data.

For low and varying wind speeds Equation (2-25) will not be valid (Murthy and Okubo, 1975). In these cases the internal mixing is considered . to be governed by local processes, i.e., the energy source is the kinetic energy fluctuations. Kullenberg (as reported by Murthy and Okubo (1975)) proposed the following relation for weak local winds:

Equation (2-26) is representative of the vertical mixing both above and below the thermocline under conditions of low wind speeds.

Tetra Tech (1975) has used the following empirical expressions for computation of the vertical eddy thermal diffusivity, K, in their three— dimensional hydrodynamic simulation of Lake Ontario.

Lake systems that are represented geometrically as a series of completely mixed horizontal slices consider advective and dispersive transport processes to occur in the vertical direction alone. Baca and Arnett (1976), in their one—dimensional hydrothermal lake model, proposed the following expression for determining the one-dimensional vertical dispersion coefficient:

The vertical eddy viscosity and eddy diffusivity concepts continue to be practical and are a popular means for simplifications of the momentum and mass conservation equations. As pointed out by Sheng and Butler (1982) and McCutcheon (1983), however, a wide variety exists among the various forms of the vertical turbulence stability functions determined empirically by various investigators, and suggest that the appropriate stability function is dependent on the type of numerical scheme used and the nature of the water body under study. The wide variation in formulations is, in part, due to the attempt to fit empirical functions determined under specific field conditions to a wider range of water body types and internal mixing phenomena. Due to the possibility of applying an empirical relationship beyond its valid limits, site—specific testing of fomulations forE and will probably be required in most model applications.

The above discussion has concentrated on the eddy diffusion concept on which many models are based. However, an alternative to this approach is the mixed layer concept which has been successfully applied by numerous investigators to predict the vertical temperature regime of lakes and reservoirs. As summarized by Harleman (1982), the mixed layer or integral energy concept involves the following: the turbulent kinetic energy (TKE) generated by the surface wind stress is transported downward and acts to mix the upper water column layer. At the interface between the upper mixed layer and the lower quiescent layer, the remaining TKE, plus any that may be locally generated by interfacial shear (minus dissipation effects), is transferred into potential energy by entraining quiescent fluid from below the interface into the mixed layer. This entrainment, in addition to any vertical advective flows, determines the thickness of the mixed layer. TKE is also produced by convective currents which occur during periods of cooling, and can contribute to the mixing process. Also, the total vertical heat balance due to surface heat flux and internal absorption must be considered in evaluating the vertical density distribution and potential energy of the water column. A discussion of the mixed layer model approach can be found in Harleman (1982), French (1983) and Ford and Stefan (1980). Models based on this approach include those by Stefan and Ford (1980), Hurley—Octavio et al. (1977), Imberger et al. (1977) and CE-QUAL-Ri (1982).

[edit] Horizontal Eddy Diffusive Transport

Generally, horizontal eddy diffusivity is several orders of magnitude greater than the vertical eddy diffusivity (see Figure 2-4). The Journal of the Fisheries Research Board of Canada (Lam.and Jacquet, 1976) reported a range of values for the horizontal diffusivity in lakes from 1O4 to 10 cm2/sec. Unlike diffusive transport In open—channel type flows, diffusion in open water, such as in lakes and oceanic regimes, cannot be effectively related to the mean flow characteristics (Watanabe et al., 1984). Oceanic or lake turbulence represents a spectrum of different eddies resulting from the breakdown of large-scale circulations in shore zones .and by wind and wave induced circulations. Attempts to analyze this.phenomenon have demonstrated that the horizontal diffusive transport, Dh depends on the length scale I of the phenomenon. The most widely used formula is the four- thirds power law:

Useful summaries of lake and ocean diffusion data are given by Yudel son 1967), Okubo (1968) and Osmidov (1958). Okubo and Osmidov (1970) have graphed the empirical relationship between the horizontal eddy diffusivity and the length scale, as shown in Figure 2-5. According to Figure 2-5:

A comprehensive collection of diffusion data in the ocean was presented by Okubo (1971), who proposed as best fit to all the data the relation:

which is graphed in Figure 2-6. According to Chrstodoulou etal. (1976), the four-thirds law seems theoretically and experimentally acceptable for expressing the horizontal eddy diffusivity in large lakes and in the ocean, providing the length scales of interest are not of the order of the size of the energy containing eddies. In addition, the four-thirds law is not fully acceptable near the shore, due to the shoreline and bottom interference.

Two examples of the use of Equation (2-29) in lake models are in Lam and Jacquet (1976) and Lick etal. (1976). Lam and Jacquet obtained the following formulation for the horizontal eddy diffusivity for lakes, based on experimental results:

As reported by Lam and Jacquet, for a grid size l,rger than 20 km, the diffusivity is expected to be essentially constant (106 cm2/sec).

Lick (1976) used a similar formulation after Osmidov (1968), Stommel (1949), Orlob (1959), Okubo (1971) and Csanady (1973):

Observations by Lick indicated values of 104 to 105 cm2/sec for Dh for the overall circulation in the Great Lakes with smaller values indicated in the near-shore regions.

The above relationships can be used as a general guide to evaluate the horizontal diffusivitles in a numerical model, where the grid size may be reg.arded as the approximate length scale of diffusion. However, as pointed out by Murthy and Okubo (1977): (1) the data upon which these empirical relations are obtained do not represent diffusion under severe weather conditions, and thus may include a bias towards relatively mild conditions; (2) the horizontal diffusivity can vary (depending primarily upon the environmental conditions) by an order of magnitude for the same length scale of diffusion; (3) the definition of the length scale of diffusion for the horizontal diffusivity is somewhat arbitrary; and (4) the horizontal diffusivity varies by an order of magnitude between the upper and lower layers of oceans and deep lakes. Thus, to develop reliable three- dimensional models the scale and stability dependence of eddy diffusivities and the large variability of the magnitude of the eddy diffusivity with depth and environmental factors (wind, waves, inflows, etc.) must somehow be Incorporated Into the models.

The formulations for horizontal eddy diffusivity discussed above are generally representative of empirical (physical) diffusion behavior and are most compatible with a three—dimensional approach. As previously discussed, horizontal dispersion is the “effective diffusion” that occurs in twodimensional mass transport equations that have been integrated over the depth. Thus the horizontal dispersion must account for both horizontal eddy diffusivity due to horizontal turbulence and concentration gradients, as well as the effective spreading caused by velocity and concentration variations over the vertical. In addition, any simplifications in the velocity field used in modeling must also be accounted for in the dispersion coefficients. The less detailed the flow field is modeled, the larger the dispersion coefficient needs to be to provide for the spreading that would occur undr the actual circulation (Christodoulou and Pearce, 1975). Therefore, the dispersion coefficients are characteristic not only of the flow conditions to be simulated, but more significantly of theway the process is modeled. Hence these coefficients are model-dependent and difficult to quantify in any general, theoretical manner. For example, many two-dimensional models use a constant dispersion coefficient over the whole model domain as well as over time despite the fact that dispersion changes both spatially and temporally as the circulation features change. An example of a model that uses constant dispersion coefficients is Christodoulou et al. (1976).

One two-dimensional model which utilizes variable dispersion coefficients (velocity dependent) In time and space, is the finite difference model by Taylor and Pagenkopf (1981). They utilize Elder’s (1959) relationship for anisotropic flow where the dispersion of a substance is proportional to the friction velocity, u, and the water depth, ii, as follows:

The above relationship is Incorporated Into the two dimensional mass conservation equation resulting In an anisotropic mixing process which calculates a dispersion coefficient at each time step and node as a function of the instantaneous flow conditions. The expressions used for the dispersion coefficients in the model are as follows:

The above model has been successfully tested agai nst dye diffusion experiments in Flushing-Bay, New York, and in Community Harbor, Sau di Arabia (Pagenkopf and Taylor (1985); Taylor and Pagenkop.f (1981)). A two-dimensional, finite element water quality model was developed by Chen etal. (1979), based on the earlier model by Christodoulou et al. (1976). They provided for flow-dependent anisotropic dispersion coefficients by using the following relationships:

Whether the two-dimensional model in question utilizes constant or flow-dependent dispersion coefficients, the dispersion mechanism is usually somewhat dependent on factors typically beyond user control , such as numerical instabilities and grid size averaging effects. It is therefore stressed that any application of a two-dimensional water quality model be verified either through site-specific salinity or dye tracer data. Naturally, when performing field tracer experiments the time and length scales of the field phenomenon should be compatible with the time and length scales to be represented in the model . For example, a dye study lasting only a few hours is not valid for verification of a model using a daily computational time step. Similarly, a dye study confined to a small portion of a large lake or estuary will not allow for verification of the model over the entire system.

[edit] Longjtudinal Dispersive Transport in Estuaries

As previously discussed, longitudinal -dispersion is the "effective diffusion" that occurs in one-dimensional mass transport equations that have been integrated over the cross sectional area perpendicular to flow. This one—dimensional approach to modeling has often been applied to tidal and nontidal rivers, and to estuaries.

The magnitude of the one-dimensional dispersion coefficient in estuaries and tidal rivers is determined in part by the time siale over which the simulation is performed. The time scale specifies the interval over which quantities that generally change instantaneously, such as tidal current, are averaged. For shorter time scales the simulated hydrodynamics and therefore water quality relationships are resolved in greater detail and hence, in such models, smaller dispersion coefficients are needed than in those which, for example, average hydrodynamics over a tidal cycle.

The magnitude of the dispersion coefficient can also be expected to change as a function of location within an estuary. Since the one— dimensional dispersion coefficient is the result of spatial averaging over a cross section perpendicular to flow, the greater the deviation between actual velocity and the area-averaged velocity, and between actual constftuent concentrations and area-averaged concentrations, the larger will be the dispersion coefficient. These deviations are usually largest near the mouths of estuaries due to density gradients set up by the interface between fresh and saline water. Strong tidal currents may also result in large dispersion coefficients.

Because of the time scale and location dependency of the dispersion coefficient, it is convenient to divide the discussion of dispersion into time varying and tidally averaged time expressions, and then to subdivide these according to estuarine location, i.e, the salinity intrusion region and the freshwater tidal region. The salinity intrusion region is that portion of the estuary where a longitudinal salinity gradient exists. The location of the line of demarcation between the salinity intrusion region and the freshwater tidal region varies throughout the tidal cycle, and also depends on the volume of freshwater discharge. It should also be noted that the freshwater tidal region can contain saline water, if the water is of uniform density throughout the region (TRACOR, 1971). There is at present no analytical method for predicting dispersion in the salinity intrusion region of estuaries: However, because of the presence of a conservative constituent (salinity), empirical measurements are easily performed. In the freshwater tidal region, analytical expressions have been developed, while empirical measurements become more difficult due to the lack of a naturally occurring conservative tracer. Empirical measurements can alternatively be based, however, on dye release experiments.

[edit] Time Varying Longitudinal Dispersion

A model which is not averaged over the tidal cycle is more capable of representing the mixing phenomena since it represents the time varying advection in greater detail. However, the averaging effects of spatial velocity gradients (shear) and density gradients must still be accounted for. The specification of longitudinal dispersion coefficients is closely associated with the type of mathematical techniques used in a given model. Most of the model developments for one-dimensional representation of estuaries has occurred in the early 1970’s, and the most prominent techniques are summarized below.

The "link-node" or network model developed originally by WRE (1972) and commonly known as the Dynamic Estuary Model (OEM) used the basic work of Feigner and Harris (1970) to describe the numerical dispersion in the constant density region of an estuary:

There exists no corresponding formulation for the longitudinal dispersion coefficient in the salinity intrusion regions of estuaries. Rather, a careful calibration procedure is required using available salinity data to prescribe the appropriate dispersion coefficients. Obviously, this approach somewhat restricts the predictive nature of such models since a substantial amount of empirical data is necessary for proper model appl ication.

Similar versions of the DEM exist in one form or another. Not all versions, however, include the option for specification of longitudinal dispersion. This stns from the fact that considerable numerical dispersion occurs in the DEM from the first order, explicit, finite difference treatment of the advective transport terms. Feigner and Harris (1970) gave some comparisons of different weightings of the first order differencing in terms of trade-offs between numerical mixing, accuracy, and stability. Work on this problem has been done by Bella and Grenney (1970) and a numerical estimate of this dispersion can be given by the following equation:

where v represents the weighting coefficient assigned to the concentrations of two adjacent nodes.

This equation shows that the numerical dispersion is a function of Δx, Δt, and the velocity, V, which is a function of location and time. This equation Is useful for estimating the magnitude of numerical dispersion. It illustrates the lack of control that the modeler has over this phenomena in the DEM.

Daily and Harleman (1972) developed a network water quality model for estuaries which uses a finite element numerical technique. The hydraulics are coupled to the salinity through the density—gradient terms in the manner formulated by Thatcher and Harleman (1972). The high accuracy finiteelement Galerkin weighted residuals technique is relatively free of artificial numerical dispersion. The longitudinal dispersion formulation combines both the vertical shear effect and the vertical density-induced circulation effect through the following expression:

One-dimensional , time varying modeling using this expression has been performed for several estuaries, a recent example being an application (Thatcher and Harleman, 1978) to the Delaware Estuary wherein the time- varying calculations were made for a periodof an entire year in order to provide a model for testing different water management policies.

For real time simulations in the constant density region of estuaries and tidal rivers, the following expression has been proposed (TRACOR, 1971):

The determination of real time dispersion coefficients in the salinity Intrusion region requires field dafa on salinity distribution. Once the field data have been collected, the magnitudes of the dispersion coefficients can be found by fitting the solution of the salinity mass transport equation to the observed data. As reported in TRACOR (1971), this technique has been applied to the Rotterdam Waterway, an estuary of almost uniform depth and width. The longitudinal dispersion coefficient was found to be a function of x, the distance measured from the mouth (ft), as follows:

At the estuary mouth, DL was found to be 13,000 ft2/sec or 40 mi2/day (1.2 x 10 cm /sec) by using the technique described above. Under the same conditions in a constant density region, Equation (2-38) predicts DL = 175 ft2/sec, or 0.5 mi2/day (1.6 x 10 cm2/sec). This illustrates the large difference that can be expected between the real time dispersion coefficient in the salinity intrusion region of an estuary and in the constant density region. For more detailed discussions of real time longitudinal dispersion in estuaries, see Holley et al. (1970) and Fischer et al. (1979).

[edit] Steady State Longitudinal Dispersion

For tidally averaged or net nontidal flow simulations, the dispersion coefficients must somehow include the effects of oscillatory tidal mixing which has been averaged out of the hydrodynamics representation. No known general analytical expressions exist for this coefficient. Hence, it is cautioned and emphasized that steady-state dispersion coefficients must be determined based on observed data, or based on empirical equations having parameters that are determined from observed data. This limitation exists for both the constant density and salinity intrusion regions of the estuary.

In their one-dimensional tidally averaged estuary model, Johanson etal. (1977) used an empirical expression, comprised of three principal components (tidal mixing, salinity gradient, and net freshwater advective flow) for the dispersion coefficient. The relative location in an estuary where each of these factors is significant, and their relative magnitudes, are shdwn in Figure 2-7.

The expression used is:

The first term on the right side of Equation(-2-47) represents mixing brought about by the oscillatory flows associated with the ebbing and flooding of the tide. The second term represents additional mixing when longitudinal salinity gradients are present. It is noted that, in practice the above formulation requirescareful calibration using field salinity data due to the high empirical dependency of this relationship.

One common method of experimentally determining the tidally averaged dispersion coefficient is by the “fraction of freshwater method,” as explained by Officer (1976). The expression is:

DL can be calculated at any location within the estuary if the river flow, cross-sectional area, and salinity or freshwater fraction distributions are known.

The above method has certain pitfalls which are pointed out by Ward and Fischer (1971) in their analysis of such an application to the Delaware Estuary. They point out that the use of a dispersion coefficient relationship, i.e., a functional relationship of dispersion to distance, which is also directly related to the measured upstream freshwater inflow, neglects entirely the basic response of the waterbody to variations in freshwater inflow. Ward and Fischer show, for example, that it may take a period of months for the estuary to adjust to a short period change in freshwater discharge and that any dispersion coefficient relationship based on a simple correlation analyses may be seriously in error.

Hydroscience (1971) has collected values of tidally averaged dispersion coefficients for numerous estuaries, and these values are shown in Table 2-3.

In his book, Officer (1976) reviews studies performed in a number of estuaries throughout the world. He discusses the dispersion coefficients which have been determined, and a suninary of values for these estuaries is contained in Table 2-4. Many values were developed using the fraction of freshwater method just discussed. Additional values for the longitudinal dispersion coefficient have been summarized in Fischer et al. (1979).

[edit] The Lagrangian Method

The models discussed in previous sections of this chapter have all been based on the Eulerian concept of assigning velocities and concentrations to fixed points on a spatial grid. As previously discussed, the fixed grid approach tends to introduce a fictitious “numerical” dispersion into the mass transport solution since the length scale of the diffusion process is somewhat artificially imposed depending on the grid detail. To avoid such a problem, an alternative approach termed the Lagrangian method has been used by Fischer (1972), Wallis (1974), and Spaulding and Pavish (1984) for models of estuaries and tidal waters. Briefly, the Lagrangian method establishes marked volumes of water, distributed along the channel axis, which are moved along the channel at the mean flow velocity. Numerical diffusion is almost entirely eliminated, since there is no allocation of concentrations to specific grid points; rather, the “grid” is a set of moving points which represent the centers of the marked volumes. Longitudinal dispersion between marked volumes can be set according to appropriate empirical or theoretical diffusion behavior (Fischer et al., 1979). The Lagrangian method has been primarily applied to channel ized estuaries such as the Suisun Marsh (Fischer, 1977) and Bolinas Lagoon (Fischer, 1972), and more recently has been extended by Spaulding and Pavish (1984) to simulate particulate transport in three dimensions.

[edit] Dispersive Transport in Rivers

[edit] Introduction

Dispersive transport in rivers is typically, but not always, modeled using one-dimensional equation such as:

Because of the difficulty of accurately solving Equation (2-49) numerically, some researchers (e.g., Jobson, 1980a; Jobson and Rathbun, 1985) have chosen a Lagrangian approach, where the coordinate system is allowed to move with the local stream velocity. Using this approach, Equation (2-49) become:

The numerically troublesome advective term does not appear in Equation (2- 50). In generel , the equation can be solved more easily and with more accuracy than Equation (2-49).

A second method used to simulate dispersive transport in rivers is to consider lateral mixing in addition to longitudinal mixing. A typical form of the two-dimensional equation is:

Note that longitudinal dispersion coefficient, DL, in Equation (2-49) is not the same as the longitudinal diffusion coefficient,εx, in Equation (2-51),typically, DL>>εx.

[edit] Longitudinal Dispersion in Rivers

Fischer (1966, 1967a, 1967b, 1968) has performed much of the earlier research on longitudinal dispersion in natural channels. Prior to Fischer, Taylor (1954) studied dispersion in straight pipes and Elder (1959) studied dispersion in an infinitely wide open cIannel. More recently Fischer et al. (1979) and Elhadi et al. (1984) have provided a comprehensive review of dispersion processes.

Researchers have shown that Equation (2-49) is valid only after some initial mixing length, often called the Taylor length or convective period. While the convective period has been a topic of active research in the literature (e.g., Fischer, 1967a and b; McQuivey and Keefer, 1976a; Chatwin, 1980), this concept is not embodied in one-dimensionalwater quality models in general use.

Table 2-5 summarizes references on stream dispersion. The references include information from at least one of the following areas:

• methods to predict DL, typically for model applications
• methods to measure 0L from field data
• data suninaries of dispersion coefficients
• approaches used to simulate dispersion in,a non—Fickian manner.

Bansal (1971), Elhadi and Davar (1976), Elhadi et al. (1984) also provide reviews of stream dispersion. To date, the predictive capabilities of expressions for dispersion coefficients have not been thoroughly tested. However, it is known that the Taylor (1954) or Elder (1959) formulas do not accurately predict dispersion coefficients for natural streams. Glover (1964) found that dispersion coefficients in natural streams were likely .to be 10 to 40 times higher than predicted by the Taylor or Elder equations. The lateral variation In stream velocity is the primary reason for the Increased dispersion not accounted for by Taylor and Elder. Fischer (1967a) quantified the contribution of the

Image:Table 2 5c.jpg

lateral velocity variation on stream dispersion.

A number of the formulas in Table 2-5 are of the type DL/(U*H) = constant. However, several researchers, including Bansal (1971), Elhadi and Davar (1976), and Beltaos (1978a) have shown that the ratio DL/(u*H) is not a constant. Figure 2-8 shows this ratio can vary by several orders of magnitude.

Two widely used methods of predicting the longItudinal dispersion coefficients were developed by Liu (1977) and Fischer (1975) and are shown in Table 2—5. Liu showed that Fischer’s method is identical to his own when β = 0.011.

Although numerous researchers (e.g., Sabol and Nordin, 1978) have shown how to include the effects of dead zones in dispersive transport, this refinement does not yet appear to be in general use in water quality models today. In fact, some water quality models do not include dispersion at all (at least physical dispersion; numerical dispersion may be present, depending on the solution technique used).

Dispersion can be neglected in certain circumstances with very little effect on the predicted concentration distributions. Thomann (1973), Li (1972), and Ruthven (1971) have investigated the influence of dispersion. Ruthven gave a particularly simple expression for a pollutant which decays at a rate k. If

then the concentration profile will be affected by no more than 10 percent if dispersion is Ignored. Consider, for example, a decaying pollutant with k = 0.5/day in a stream where U = 1 fps and DL = 500 ft2/sec. The ratio kDL/U^2 =.003, which Indicates that dispersion can be ignored. This guideline assumes that the pollutant is being continuously released and conditions are at steady state. The basic presumption is that if the concentration gradient is small enough, the dispersive transport is also small, and

perhaps negligible. On the other hand when pollutants are spilled, concentration gradients are large and dispersion is not negligible.

Thomann (1973) investigated the importance of longitudinal dispersion in rivers that received time variable waste loadings, and therefore produced concentration gradients in the rivers. His results showed that for small rivers, dispersion may be important when the waste loads vary with periods of 7 days or less. For large rivers, dispersion was found to be important whenever the waste load was time—variable.

[edit] Lateral Dispersion In Rivers

Although two-dimensional water quality models are less widely used in rivers than one-dimensional models, lateral mixing has been the topic of considerable research. Models that simulate lateral mixing are particularly useful in wide rivers where the one dimensional approach may not be applicable. Vertical mixing is rarely simulated in river modeling because the time required for vertical mixing is usually very rapid compared to the time required for lateral mixing. Thermal plumes are an exception.

An example of a model that simulates lateral mixing in rivers is the RI VMIX model of Krlshnappan and Lau (1982). The model is particularly useful for delineating mixing zones or regulating the rate of pollutant discharge so that concentrations outside of the mixing zones are limited to allowable values.

When lateral and longitudinal mixing are both simulated, the x and y coordinates are generally assumed to continuously change to be oriented in the longitudinal and transverse directions. Although Equation (2—51) should rigorously contain metric factors (Fukuoka and Sayre, 1973) to account for these continuous changes, modelers typically assume the metric factors are unity.

Lateral mixing coefficients are usually presented in one of the following two forms:

Equation (2-52) is generally the most widely used of the two formulas. Equation (2-53) is used when the two—dimensional convective—diffusion. equation is expressed in terms of cumulative discharge (Yotsukura and Cobb, 1972).

Table 2-6 summarizes studies of transverse mixing in streams. Data from the literature are summarized in Tables 2-7 through 2-9. Table 2-9 contains values of f for use in Equation (2-53).

Elhadi et al. (1984) have recently provided a detailed review of lateral mixing in rivers. They concluded that lateral mixing coefficients can be precflcted with accuracy only In relatively straight channels.

[edit] Summary

The previous sections have provided a brief review on the treatment of dispersive transport in water quality models. This has included a discussion of vertical dispersion in lakes and estuaries, and horizontal (lateral and longitudinal) dispersion in lakes, estuaries, and rivers. It is readily seen that a wide variety of numerical formulations for dispersion exist in the literature. Formulations for dispersion coefficients tend to be model-dependent and are all based to some extent on general lack of a complete understanding of the highly complex turbulence induced mixing processes which exist In natural water bodies. In all cases, due to this model and empirical dependence, It is desirable to include a careful calibration and/or verification exercise using on—site field data for any water quality modeling application.

[edit] Surface Heat Budget

The total heat budget for a water body includes the effects of inflows (rivers, discharges), outflows, heat generated by chemical-biological reactions, heat exchange with the stream bed, and atmospheric heat exchange at the water surface. In all practicality, however, the dominant process controlling the heat budget Is the atmospheric heat exchange, which Is the focus of the following paragraphs. In addition, however, it is also important to include the proper boundary conditions for advective exchange (e.g., rivers, thermal discharges, or tidal flows) when the relative source temperature and rate of advective exchange is great enough to affect the temperature distribution of the water body.

The transfer of energy which occurs at the air-water interface is generally handled in one of two ways in river, lake, and estuary models. A simplified approach is to input temperature values directly and avoid a more complete formulation of the energy transfer phenomena. This approach is most often applied to those aquatic systems where the temperature can be readily measured. Alternatively, and quite conveniently, the various energy transfer phenomena which occur at the air-water iftterface can be considered in a heat budget formulation.

In a complete atmospheric heat budget formulation, the net external heat flux, H, is most often formul ated as an algebraic sum of several component energy fluxes (e.g., Baca and Arnett, 1976; U.S. Army Corps of Engineers, 1974; Thomann et al., 1975; Edinger and Buchak, 1978; Ryan and Harleman, 1973; TVA, 1972). A typical expression is given as:

These flux components can be calculated within the models from semi— theoretical relations, empirical equations, and basic meteorological data. Depending on the algebraic formulation used for the net heat flux term and the particular empirical expressions chosen for each component, all or some of the following meteorological data may be required: atmospheric pressure, cloud cover, wind speed and direction, wet and dry bulb air temperatures, dew point temperature, short wave solar radiation, relative humidity, water temperature, latitude, and longitude.

Estimation of the various heat flux components has been the subject of many theoretical and experimental stud.ies in the late 1960’s and early 1970s. Most of the derived equations rely heavily on empirical coefficients. These formulations have been reviewed extensively by the Tennessee Valley Authority (1972), Ryan and Harlernan (1973), Edinger et al. .(1974), and Paily et al. (1974). A summary of the most commonly used formulations in water quality models is given in the following sections.

[edit] Measurement Units

The measurement units in surface heat transfer calculations do not follow any consistent units system. For heat flux, the English system units are BTU/ft2/day. In the metric system, the units are either Kcal/m2/hr or watt/rn2 (1 watt = 1 joule/sec). The Langley (abbreviated Ly), equal to 1 cal/cm2, also persists in usage. The following conversions are useful in this section:

[edit] Net short wave Solar Radiation, Qsn

Net short wave solar radiation is the difference between the incident and reflected solar radiations, Qsr• Techniques are available and described in the aforementioned references to estimate these fluxes as a function of meteorological data. However, in order to account for the reflection, scattering, and absorption incurred by the radiation through interaction with gases, water vapor, clouds, and dust particles, a great deal of empiricism is involved and the necessary data are relatively extensive if precision is desired.

One of the most cormnon simplified formulations for net short wave solar radiation (Anderson, 1954; Ryan and HarIeman, 1973) is expressed as:

As reported by Shanahan (1984), Equation (2-56) Is an approximation in that it assumes average reflectance at the water surface and employs clear sky solar radiation. In certain circumstances atmospheric attenuation mechanisms are much greater than normal, even under cloudless conditions. For such situations, the more complex formulae described by TVA (1972) are required.

A number of methods are available for estimating the clear sky solar radiation. TVA (1972) presents a formula for as a function of the geographical location, time of year, and hour of the day. Thackston (1974) and Thompson (1975) report methods for calculating daily average values of solar radiation as a function of latitude, longitude, month, and sky cover. Hamon et al. (1954) have graphed the daily average insolation as a function of latitude, day of year and percent of possible hours of sunshine, and is given in Figure 2—9.

Lombardo (1972) represents the net short wave solar radiation, Qsn (langleys/day), with the following expression:

The WQRRS model by the U.S. Army Corps of Engineers (1974) considers the net short wave solar radiation rate - sr as a function of sun angle, cloudiness, and the level of particulates in the atmosphere. Chen and Orlob, as reported by Lombardo (1973), determine the net shor,t wave solar radiation by considering absorption and scattering in the atmosphere.

A final important note on calculation of the net short wave solar radiation regards the effects of shading from trees and banks primarily on stream systems or rivers with steep banks. Shading can significantly reduce the incoming solar radiation to the water surface, resulting in water temperatures much lower than those occurring in unobstructed areas. Jobson and Keefer (1979) present a method to account for the reduction of incoming solar radiation by prescribing geometric relations of vertical obstruction heights and stream widths for each subreach of their model of the Chattahoochee River.

[edit] Net Atmospheric Radiation, Qan

The atmospheric radiation is characterized by much longer wavelengths than solar radiation since the major emitting elements are water vapor, carbon dioxide, and ozone. The approach generally adopted to compute this flux involves the empirical determination of an overall atmospheric emissivity and the use of the Stephan-Boltzman law (Ryan and Harleman, 1973). The formula by Swinbank (1963) has been adopted by many investigators for use in various water quality models (e.g., U.S. Army Corps of Engineers, 1974; Chen and Orlob, 1975; Brocard and Harleman, 1976). This formula was believed to give reliable values of the atmospheric radiation within a probable error to +5 percent. Swinbank’s formula is:

A recent investigation by Hatfield et al. (1983) has found that the formula by Brunt (1932 gives more accurate results over a range of latitudes of 26°13’N to 47°45’N and an elevation range of -30m to + 3,342m. Brunt’s formula is:

[edit] Long Wave Radiation, Qbr

The long wave back radiation from the water surface is usually the largest of all the fluxes in the heat budget (Ryan and Harleman, 1973). Since the emissivity of a water surface (0.97) is known with good precision, this flux can be determined with accuracy as a function of the water surface temperature:

The U.S. Army Corps of Engineers (1974) uses the following linearization of Equation (2-60) to express the emitted by the water body:

In the range of 00 to 30°C, this linear function has a maximum error of less than 2.1 percent relative to Equation (2-60).

[edit] Evaporative Heat Flux, Qe

Evaporative heat loss occurs as a result of the change of state of water from a liquid to vapor, requiring sacrifice of the latent heat of vaporizati.on. The basic formulation used in all heat budget formulations (e.g., Ryan and Harleman, 1973; U.S. Army Corps of Engineers, 1974; Chen and Orlob, 1975; Lombardo, 1972) is:

The general expression for evaporation from a natural water surface is usually written as:

Various approaches have been used to evaluate the above expression. In a very simplified approach, the empirical coefficient, a, has often been taken to be zero, while b ranges from 1 x 10 to 5 x 10 (U.S. Army Corps of Engineers, 1974). The value of e5 is a nonlinear function of the surface water temperature. However esscan be estimated in a piecewise linear fashion as follows:

A more convenient formula for the saturation vapor pressure, es, is presented by Thackston (1974) as follows:

The standard error of prediction of Equation (2-55) is reported by Thackston (1974) to be 0.00335.

A large number of evaporation formula exist for a natural water surface, as demonstrated in Table 2-12 (Ryan and Harleman, 1973). Detailed comparisons of these formulae by the above authors showed that the discrepancies between these formulae were not significant. Both Ryan and Harleman (1973), and TVA (1968) recommend the use of the Lake Hefner evaporation formula developed by Marciano and Harbeck (1954), which has the best data base, and has been shown to perform satisfactorily for other water bodies. The Lake Hefner formula is written as:

It is Important to note that the Lake Hefner formula was developed for lakes and may not be universally valid for streams or open channels due to physical blockage of the wind by trees, banks, etc.; and due to differences in the surface turbulence which affects the liquid film aspects of evaporation (McCutcheon, 1982). Jobson developed a modified evaporation formula which was used in temperature modeling of the San Diego Aqueduct (Jobson, 1980) and the Chattahoochee River (Jobson and Keefer, 1981). This formula is written as:

It is noted that the wind speed function of Equation (2-67) was reduced by 30 percent during calibration of the temperature model for the Chattahoochee River (McCutcheon, 1982). The original Equation (2-67) was developed for the San Diego Aqueduct which represented substantially different climactic and exposure conditions than for the Chattahoochee River. McCutcheon (1982) notes that the wind speed function Is a catchall term that must compensate for a number of difficulties which Include, in part:

• Numerical dispersion in some models.
• Inaccuracies in the measurement and/or calculation of wind speed, solar and long-wave radiation, air temperature, cloud cover, and relative humidity.
• Effects of wind direction, fetch, channel width, sinuosity, bank and tree height.
• effects of depth, turbulence, and lateral velocity distribution.
• Stability of the air moving over the stream.

[edit] Convective Heat Flux, Qc

Convective heat is transferred between air and water by conduction and transported away from (or toward) the air-water interface by convection associated with the moving air mass. The convective heat flux is related to the evaporative heat flux, Qe, through the Bowen ratio:

The above formulation is used in the surface heat transfer budget of several models (e.g., U.S. Army Corps of Engineers, 1974; Brocard and Harleman, 1976).

[edit] Equilibrium Temperature and Linearization

The preceding paragraphs present methods for estimating the magnitudes of the various components of heat transfer through the water surface. Several of these components are nonlinear functions of the surface water temperature, Ts. Thus, they are most appropriately used in transient water quality simulations where the need to predict temperature variations is on the time scale of minutes or hours. However, for long term water quality simulations or for steady state simulations, it is more economical to use a linearized approach to heat transfer. As developed by Edinger and Geyer (1965), and reported by Ryan and Harleman (1973), this approach involves two concepts, that of equilibrium temperature, Te, and surface heat exchange, K, where H can now be written as:

The equilibrium temperature, Te, is defined as that water surface temperature which, for a given set of meteorological conditions, causes the surface heat flux H, to equal zero. The surface heat exchange coefficient, K, is defined to give the incremental change of net heat exchange induced by an incremental change of water surface temperature. It varies with the surface temperature and thus should be recalculated as the water temperature changes.

[edit] Equilibrium temperature, Te

The equilibrium temperature Te is the temperature toward which every water body at the site will tend, and is useful because it is dependent solely upon meteorological variables at a given site. A water body at a surface temperature, Tw less than Te, will have a net heat input and thus will tend to increase its temperature. The opposite is true if Tw > Te. Thus, the equilibrium temperature embodies all the external influences upon ambient temperatures.

Certain formulations for the equilibrium temperature have been developed which require an iterative or trial and error solution approach (Ryan and Harleman, 1973). An approximate formula for obtaining Te has been developed by Brady et al. (1969) which has been shown to yield fairly accurate results:

The expression for β is written as:

[edit] Surface Heat Exchange Coefficient, K

The surface heat exchange coefficient, K, relates the net heat transfer rate to changes In water surface temperature. An expression for K developed by Brady et al. (1969), (and reported by Ryan and Harleman, 1973) is:

Charts giving K as a function of water surface temperature and wind speed are given by Ryan and Stoizenbach (1972), assuming an average relative humidity of 75 percent. Shanahan (1984) presents a calculation procedure to determine Te and K from average meteorological data.

[edit] Heat Exchange with Stream Bed

For most lakes, estuaries, and deep rivers, the thermal flux through the bottom is insignificant. However, as reported by Jobson (1980) and Jobson and Keefer (1979), the bed conduction term may be significant in determining the diurnal variation of temperatures in water bodies with depths of 10 ft (3m) or less. Jobson (1977) presents a procedure for accounting for bed conduction which does not require temperature measurements within the bed. Rather, the procedure estimates the heat exchange based on the gross thermal properties of the bed, including the thermal diffusivity and heat storage capacity. The inclusion of this method improved dynamic temperature simulation on the San Diego Aqueduct and the Chattahoochee River.

[edit] Summary

The previous section has presented a brief summary of the most frequently used formulations for surface heat exchange in numerical water quality models. These formulations are widely used and have been shown to work quite well within the normal range of meteorological and surface water conditions, provided a reasonably complete data base is available on meteorological conditions at the site of interest. Meteorological data requirements include atmospheric pressure, cloud cover, and at a known surface elevation: wind speed and direction, relative humidity, and wet and dry bulb air temperatures. Shanahan (1984) presents a useful summary of meteorological data requirements for surface heat exchange computations.

[edit] References

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